To add an element to a vector of integers named numbers at the next available position in the vector, you would use numbers.pushBack(newValue); numbers[numbers.size( )+1] = newValue; numbers.push_back(newValue); numbers = newValue;

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To add an element to a `std::vector` named `numbers` at the next available position, you should use the `push_back` member function. Here's a breakdown of the options you provided:

1. **`numbers.pushBack(newValue);`**
   - **Incorrect**: The correct function name uses an underscore and a lowercase 'b'. C++ is case-sensitive, so `pushBack` is not recognized.

2. **`numbers[numbers.size() + 1] = newValue;`**
   - **Incorrect**: This attempts to access an index that is out of bounds. Since vector indices are zero-based, the next available position is `numbers.size()`. However, directly assigning to this index without resizing the vector first leads to undefined behavior.

3. **`numbers.push_back(newValue);`**
   - **Correct**: This is the proper way to add a new element to the end of the vector. The `push_back` function appends the element `newValue` to the vector, handling any necessary memory management internally.

4. **`numbers = newValue;`**
   - **Incorrect**: This attempts to assign an integer (`newValue`) directly to a vector, which is a type mismatch and will result in a compilation error.

### Correct Usage Example

```cpp
#include

int main() {
    std::vector numbers;
    int newValue = 42;

// Correct way to add a new element
    numbers.push_back(newValue);

return 0;
}
```

### Additional Notes

- **`push_back` vs. `emplace_back`**: While `push_back` is suitable for adding existing objects to the vector, `emplace_back` can be more efficient in some cases as it constructs the element in place.
  
  ```cpp
  numbers.emplace_back(newValue);
  ```

- **Resizing Before Assignment**: If you prefer to use the subscript operator (`[]`), ensure that the vector has enough space by resizing it first.

```cpp
  numbers.resize(numbers.size() + 1);
  numbers[numbers.size() - 1] = newValue;
  ```

However, using `push_back` is generally more straightforward and less error-prone for simply appending elements.

### Summary

Use `numbers.push_back(newValue);` to correctly add a new element to the end of the vector.
Use `numbers.push_back(newValue);` to add a new element to the vector.

To add an element to a vector of integers named numbers at the next available position in the vector, you would use numbers.pushBack(newValue); numbers[numbers.size( )+1] = newValue; numbers.push_back(newValue); numbers = newValue;

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